Canonical Jahn-Teller Distortion Notations: Understanding Structural-Electronic Interplays in e1g Perovskites

Canonical Jahn-Teller Distortion Notations

Understanding Structural-Electronic Interplays in e

1

g

Perovskites

Michael-Marcus Schmitt,Yajun Zhang, Alain Mercy, Eric Bousquet, & Philippe Ghosez

2019 Workshop on Transition Metal Oxide Physics Hotel Bristol, Stresa, Italy

16th - 19th October 2019

Physique Théorique des Matériaux, Q-MAT, CESAM, Université de Liège

Outline

Canonical Jahn-Teller Distortion Notations Application I: Bulk Ground State of LaMnO3

Application II: [001] Epitaxial Strain Phase Diagram of CaFeO3

Canonical Jahn-Teller

Distortion Notations

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck, J. H. The Journal of Chemical Physics 7.1 (1939): 72-84.

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck, J. H. The Journal of Chemical Physics 7.1 (1939): 72-84.

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q−2,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck, J. H. The Journal of Chemical Physics 7.1 (1939): 72-84.

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q2−,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q2−,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Van Vleck, J. H. The Journal of Chemical Physics 7.1 (1939): 72-84.

Van Vleck: The octahedral Complex MX

6

John Hasbrouck Van Vleck

M

Crystal Fi eld Split ting

X

Oh

eg⊗eg=a1g+eg Q1 a1g Q2 Q3 eg t2g⊗t2g=a_1g+e_g+t_2g Q4,Q₅,Q₆ t2g

Many different Notations for this in the literature!

Chemists

Qθ,Q

O’Brien, M. C. & Chancey, C. Am. J. Phys., 1993, 61, 688-697 Labels of Irreducible Representation

M2+, M3+, R3-, R3+, R4-...

Carpenter, M. A. & Howard, C. J. Acta Crystallogr. B., 2009, 65, 134-146 Solid State Physicists

QM

1 ,Q1R,Q2+,Q2−,MJT,R_JT,Qx,Qz,Qx_R,Q_Rz... He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Varignon, J.; Bristowe, N. C. et al.; Sci. Rep., 2015, 5, 15364

Varignon, J.; Bristowe, N. C. & Ghosez, P. E Phys. Rev. Lett., 2016, 116, 057602 Ederer, C.; Lin, C. & Millis, A. J. Phys. Rev. B, 2007, 76, 155105

Perovskite Structure

Same Individual Distortion

Through Different Cooperative Arrangements

A Canonical Notation!

i = Vlecks Numbering ~q = q-vector in Cubic BZ α= orientation (x,y,z) Q_i·eiq~R

Q

~

q

iα

1 2 3 4 Γ = (0, 0, 0) = Strain X = (1 2,0, 0) M = (1 2,12,0) R = (1 2,12,12)

He, Z. & Millis, A. J. Phys. Rev. B, 2015, 91, 195138 Kanamori, J. J. Appl. Phys., 1960, 31, 14-23 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

1

-Modes and Strains

Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

1

-Modes and Strains

RNiO3

Mercy, A.et al., Nat. Commun., 2017, 8, 1677 Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

1

-Modes and Strains

Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

1

-Modes and Strains

Q

1

-Modes and Strains

Park, S. Y. et al. Phys. Rev. Lett., 2017, 118, 087602

Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

2

/

Q

3

-Modes and Strains

Q

2

/

Q

3

-Modes and Strains

RMnO3

QM 2z & Q3zΓ

KCuF3

QR 2z & QΓ3z

KCrF3

QR 2z & QΓ3z

Schmitt, M. M., et al, arXiv:1909.06287 (2019)

Q

4

,

Q

5

,

Q

6

-Modes and Strains

0.84 0.86 0.88 0.9 0.92 0.94 Goldschmidt Factor t -0.02 0 0.02 0.04 Am p. Y Lu Yb Lu Yb Tm Er Ho C e TbTm Er Ho Tb Nd Pr Ce La Y 0.82 0.84 0.86 0.88 0.9 0.92 Y Gd Sm Nd La a) RVO₃ b) RTiO₃ QΓ3 Q4zΓ

Martínez-Lope, M. J. et al. Inorg. Chem., 2008, 47, 2634-264 Komarek, A. C. et al. Phys. Rev. B, 2007, 75, 224402

CJTN - Strain Basis

a

1g QΓ 1

e

g QΓ 2α& Q3αΓ

t

2g QΓ 4x,y,z ˆ =    xx xy xz _yy _yz sym zz   =(ˆ 1, 2, 3, 4, 5, 6)

Application I: Bulk Ground

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

JTD In LaMnO

3 GS Pnma a−_a−_c+ + JTD Metallic O-Phase 750K < T < 1200K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Mn3+_=d4

→

Ins. O’-Phase T < TJT =750K - AFM-A T < 140K

QM

2z QΓ3z QΓ4z

How do structural and electronic degrees of freedom interact ?

Let’s ask DFT!

K-Mesh 14x14x14 - Ecut =600eV Exc = PBEsol + (U|J)

U=5.5 eV J=1.5 eV Ground State Structure Band Gap (1.1 eV - Exp 1.1 - 1.9 eV)

Anistropy of Dielectric Tensor Magnetic Exchange Constants

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

Rivero, P., et al., Phys. Rev. B 93.9 (2016)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−Q_2zM F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

Rivero, P., et al., Phys. Rev. B 93.9 (2016)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−Q_2zM F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β2<0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−Q_2zM F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β₃| > |β₄|

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

Rivero, P., et al., Phys. Rev. B 93.9 (2016)

ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic F ∝ αelQM2z+ β1QM2z2 F ∝ αelQM2z+ β1Q2zM2+ β2φ2QM_2z2 β <0 F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−Q_2zM F ∝ αelQM2z+ β1QM2z2+ β2φ2QM_2z2+ γ1AXφ−_QM 2z+ β3Q3zΓQ2zM2+ β4Q3zΓ2QM2z2 |β | > |β |

Tetragonal Strain Q

Γ 3z

controls MO !

In FM thin films Q

Γ 3z

u 0

Marton, Z., et al., J. Cryst. Growth 312.20 (2010) Hou, Y. S., et al. Phys. Rev. B 89.6 (2014) Roqueta, J., et al., Cryst Growth & Des 15.11 (2015) Vila-Fungueiriño, J.M. et al. ACS Appl. Mater. Interfaces 7.9 (2015)

Application II: [001] Epitaxial

Strain Phase Diagram of

GS of CaFeO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n Metallic Pnma T > 290K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Fe4+_=d4

→

Ins. P21/n T < T_MIT u 290K AFM T < 120K QR 1

Effect of [001] epitaxial strain?

→

Q

Γ

GS of CaFeO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n Metallic Pnma T > 290K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Fe4+_=d4

→

Ins. P21/n T < T_MIT u 290K AFM T < 120K QR 1

Effect of [001] epitaxial strain?

→

Q

Γ

3z

6=

0

GS of CaFeO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n Metallic Pnma T > 290K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Fe4+_=d4

→

Ins. P21/n T < T_MIT u 290K AFM T < 120K QR 1

Effect of [001] epitaxial strain?

→

Q

Γ

GS of CaFeO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n Metallic Pnma T > 290K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Fe4+_=d4

→

Ins. P21/n T < T_MIT u 290K AFM T < 120K QR 1

Effect of [001] epitaxial strain?

→

Q

Γ

3z

6=

0

GS of CaFeO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n Metallic Pnma T > 290K ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Fe4+_=d4

→

Ins. P21/n T < T_MIT u 290K AFM T < 120K QR 1

Effect of [001] epitaxial strain?

→

Q

Γ

[001] Epitaxial Strain Tuning: From Charge- to

Orbital-Order

F ∝ β3QΓ3zQM 2 2z + β4QΓ3z2QM 2 2z |β3| >> |β4| F ∝ β5Q3zΓ2QR 2 1 β₅>0 Experimental

Paul C. Rogge et al Phys. Rev. Materials 2, 015002 (2018)

Zhang, Schmitt et al. Phys. Rev. B 98, 081108(R), (2018)

[001] Epitaxial Strain Tuning: From Charge- to

Orbital-Order

F ∝ β3QΓ3zQM 2 2z + β4QΓ3z2QM 2 2z |β3| >> |β4| F ∝ β5Q3zΓ2QR 2 1 β₅>0 Experimental

Paul C. Rogge et al Phys. Rev. Materials 2, 015002 (2018)

[001] Epitaxial Strain Tuning: From Charge- to

Orbital-Order

F ∝ β3QΓ3zQM 2 2z + β4Q3zΓ2QM 2 2z |β3| >> |β4| F ∝ β5Q3zΓ2QR 2 1 β₅>0 Experimental

Paul C. Rogge et al Phys. Rev. Materials 2, 015002 (2018)

Zhang, Schmitt et al. Phys. Rev. B 98, 081108(R), (2018)

[001] Epitaxial Strain Tuning: From Charge- to

Orbital-Order

F ∝ β3QΓ3zQM 2 2z + β4Q3zΓ2QM 2 2z |β3| >> |β4| F ∝ β5Q3zΓ2QR 2 1 β₅>0 Experimental

Application III: [001] vs [111]

Epitaxial Strain in NdNiO

3

GS of NdNiO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n T > 200K Metallic Pnma ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Ni3+_=d7

→

T < TMIT u 200K Ins. P21/n AFM T < TN u 200K QR 1

GS of NdNiO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n T > 200K Metallic Pnma ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Ni3+_=d7

→

T < TMIT u 200K Ins. P21/n AFM T < TN u 200K QR 1

Effect of [001] vs [111] epitaxial strain?

GS of NdNiO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n T > 200K Metallic Pnma ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Ni3+_=d7

→

T < TMIT u 200K Ins. P21/n AFM T < TN u 200K QR 1

GS of NdNiO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n T > 200K Metallic Pnma ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Ni3+_=d7

→

T < TMIT u 200K Ins. P21/n AFM T < TN u 200K QR 1

Effect of [001] vs [111] epitaxial strain?

GS of NdNiO

3 GS Pnma a−_a−_c+ + QR 1 →P21/n T > 200K Metallic Pnma ϕz+ AX x y z x y z b a c b a c ϕ -xy x Cubic Ni3+_=d7

→

T < TMIT u 200K Ins. P21/n AFM T < TN u 200K QR 1

[001] vs. [111] Epitaxial Strain

Catalano, S., et al. APL materials 3.6 (2015)

[001] vs. [111] Epitaxial Strain

[001] vs. [111] Epitaxial Strain

Catalano, S., et al. APL materials 3.6 (2015)

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

3z

=

0 while Q

φ

and Q

Γ1

same as on [001] NGO

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

3z

=

0 while Q

φ

and Q

Γ1

same as on [001] NGO

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

[001] vs. [111] Epitaxial Strain

QΓ

1 6=0 & Q3zΓ 6=0 QΓ1 6=0 & QΓ4x,y,z6=0

QΓ 4x,y,z =0 Q3zΓ =0 NNO NNO /NGO[001]pc /NGO[111]pc QΓ 1 0.014 0.014 QΓ 3z -0.029 0.000 QΓ 4z 0.009 0.026 QΓ 4x,y 0.000 0.016

Tetragonal Strain Q

Γ 3z

hinders MIT

Q

Γ

3z

competes with Q

1Γ

and increased rotations

On NGO [111] Q

Γ

3z

=

0 while Q

φ

and Q

Γ1

same as on [001] NGO

Conclusions

• Canonical Jahn-Teller Distortion Notations Q~q

iα for anaylizing

Perovskite Materials

→Schmitt, M. M., et al, arXiv:1909.06287 (2019) • Tetragonal Strain QΓ

3αcooperative with orbital-ordered

(Jahn-Teller distorted) phases in e1

g perovskites

• Tetragonal Strain QΓ

3αanti-cooperative with charge-ordered

(QR

1-distorted) phases in e1g perovskites

• Strain and Rotation state in thin film can be engineered by choice of substrate:

→Space-Group of Substrate (Rotations/No-Rotations) →Surface of Substrate [001],[011],[111]..

Thank you for your attention!

Questions?

Ferromagnetic LaMnO

3

on SrTiO

3

c c

Roqueta J.,et al. Cryst. Growth Des. 15.11 (2015)

STO a=b=c 3.905 Å c0 a0 b0 a b c LMO-STO LMO-Bulk P-1 Pnma FM AFM-A QΓ 3 -0.005 -0.04 QΓ 4z -0.018 -0.036 QM 2 (Å) 0.117 0.19 QR 3 (Å) 0.077 -φ+z (Å) 0.44 0.49 φ−xy(Å) 0.62 0.65 AX(Å) 0.26 0.33

Ferromagnetic LaMnO

3

on SrTiO

3

c c

Roqueta J.,et al. Cryst. Growth Des. 15.11 (2015)

STO a=b=c 3.905 Å c0 a0 b0 a b c LMO-STO LMO-Bulk P-1 Pnma FM AFM-A QΓ 3 -0.005 -0.04 QΓ 4z -0.018 -0.036 QM 2 (Å) 0.117 0.19 QR 3 (Å) 0.077 -φ+z (Å) 0.44 0.49 φ−xy(Å) 0.62 0.65 AX(Å) 0.26 0.33

Ferromagnetic LaMnO

3

on SrTiO

3

c c

Roqueta J.,et al. Cryst. Growth Des. 15.11 (2015)

STO a=b=c 3.905 Å c0 a0 b0 a b c LMO-STO LMO-Bulk P-1 Pnma FM AFM-A QΓ 3 -0.005 -0.04 QΓ 4z -0.018 -0.036 QM 2 (Å) 0.117 0.19 QR 3 (Å) 0.077 -φ+z (Å) 0.44 0.49 φ−xy(Å) 0.62 0.65 AX(Å) 0.26 0.33

Ferromagnetic LaMnO

3

on SrTiO

3

c c

Roqueta J.,et al. Cryst. Growth Des. 15.11 (2015)

STO a=b=c 3.905 Å c0 a0 b0 a b c LMO-STO LMO-Bulk P-1 Pnma FM AFM-A QΓ 3 -0.005 -0.04 QΓ 4z -0.018 -0.036 QM 2 (Å) 0.117 0.19 QR 3 (Å) 0.077 -φ+z (Å) 0.44 0.49 φ−xy(Å) 0.62 0.65 AX(Å) 0.26 0.33

Ferromagnetic LaMnO

3

on SrTiO

3

c c

Roqueta J.,et al. Cryst. Growth Des. 15.11 (2015)

STO a=b=c 3.905 Å c0 a0 b0 a b c LMO-STO LMO-Bulk P-1 Pnma FM AFM-A QΓ 3 -0.005 -0.04 QΓ 4z -0.018 -0.036 QM 2 (Å) 0.117 0.19 QR 3 (Å) 0.077 -φ+z (Å) 0.44 0.49 φ−xy(Å) 0.62 0.65 AX(Å) 0.26 0.33

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R 0% a-a-c+ 50% a-a-c+ 100% a-a-c+ ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M -80 -60 -40 -20 0 20 40 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R ∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R

Q

R 3

JTD

∆ Ε [meV] -2 -1 0 1 2 No Q2M With Q2M -80 -60 -40 -20 0 20 40 -2 -1 0 1 2 Amp. Q3R -2 0 2 -2 0 2 -2 0 2 E - E F [eV] -2 0 2 Γ X S Y Γ Z U R T Z Y T U X S R -2 0 2 Cubic a-_a-_c+ a-_a-_c+ + Q3R a-_a-_c+ + Q2M a-_a-_c+ +Q2M+Q3R

Strain Engineering in LaMnO

3 LaMnO3 Substrate -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c 0.25 0.5 0.75 Amp. [Å] ϕ+ z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -0.04 -0.02 0 0.02 Amp. Q₃Γ Q₂Γ 0.25 0.5 0.75 Amp. [Å] ϕ+ z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 0 0.1 0.2 Amp. [Å] QR₃ QM₂ Pnma c AFM-A P-1 c FM 0.25 0.5 0.75 1 1.25 1.5 EGap [eV] FM cAFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 4 6 8 10 ε∞ εaa ε_bb ε_cc Pnma c AFM-A P-1 c FM

Strain Engineering in LaMnO

3 LaMnO3 Substrate -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c 0.25 0.5 0.75 Amp. [Å] ϕ+ z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -0.04 -0.02 0 0.02 Amp. Q₃Γ Q₂Γ 0.25 0.5 0.75 Amp. [Å] ϕ+ z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 0 0.1 0.2 Amp. [Å] QR₃ QM₂ Pnma c AFM-A P-1 c FM 0.25 0.5 0.75 1 1.25 1.5 EGap [eV] FM cAFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 4 6 8 10 ε∞ εaa ε_bb ε_cc Pnma c AFM-A P-1 c FM

Strain Engineering in LaMnO

3 LaMnO3 Substrate -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c 0.25 0.5 0.75 Amp. [Å] ϕ+_z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -0.04 -0.02 0 0.02 Amp. Q₃Γ Q₂Γ 0.25 0.5 0.75 Amp. [Å] ϕ+ z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 0 0.1 0.2 Amp. [Å] QR₃ QM₂ Pnma c AFM-A P-1 c FM 0.25 0.5 0.75 1 1.25 1.5 EGap [eV] FM cAFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 4 6 8 10 ε∞ εaa ε_bb ε_cc Pnma c AFM-A P-1 c FM

Strain Engineering in LaMnO

3 LaMnO3 Substrate -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c 0.25 0.5 0.75 Amp. [Å] ϕ+_z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -0.04 -0.02 0 0.02 Amp. Q₃Γ Q₂Γ 0.25 0.5 0.75 Amp. [Å] ϕ+_z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 Amp. [Å] QR₃ QM₂ Pnma c AFM-A P-1 c FM 0.25 0.5 0.75 1 1.25 1.5 EGap [eV] FM cAFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 4 6 8 10 ε∞ εaa ε_bb ε_cc Pnma c AFM-A P-1 c FM

Strain Engineering in LaMnO

3 LaMnO3 Substrate -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c 0.25 0.5 0.75 Amp. [Å] ϕ+_z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] Pnma c AFM-A P-1 c FM -20 0 20 40 60 E/fu [meV] 3.875 3.895 3.915 3.935 3.955 3.975 3.995 Lattice Constant a₀[Å] FM c AFM-A c -0.04 -0.02 0 0.02 Amp. Q₃Γ Q₂Γ 0.25 0.5 0.75 Amp. [Å] ϕ+_z ϕ− xy A_X -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 0 0.1 0.2 Amp. [Å] QR₃ QM₂ Pnma c AFM-A P-1 c FM 0.25 0.5 0.75 1 1.25 1.5 EGap [eV] FM cAFM-A c -1.5 -1 -0.5 0 0.5 1 1.5 Epitaxial Strain [%] 4 6 8 10 ε∞ εaa ε_bb ε_cc Pnma c AFM-A P-1 c FM

Modes and Strains across T

MIT

LaMnO

3 300 400 500 600 700 800 900 1000 T (K) -0.04 -0.03 -0.02 -0.01 0 0.01 A m p. 300 400 500 600 700 800 900 1000 T (K) 0 0.1 0.2 0.3 0.4 0.5 0.6 A m p. (Å ) A_X Q2M Q3Γ Q4zΓ ϕxy -ϕ+z Modes Strains O' O O' O

Modes and Strains across T

MIT

LaMnO

3 300 400 500 600 700 800 900 1000 T (K) -0.04 -0.03 -0.02 -0.01 0 0.01 A m p. 300 400 500 600 700 800 900 1000 T (K) 0 0.1 0.2 0.3 0.4 0.5 0.6 A m p. (Å ) A_X Q2M Q3Γ Q4zΓ ϕxy -ϕ+z Modes Strains O' O O' O

Full Potential Energy Surface LaMnO

3 -1 0 1 -2 -1,5 -1 -0,5 0 0,5 Energy [eV] -1 0 1 AFM-A FM -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 0,75 1 -1,65 -1,6 -1 0 1 0,75 1 -1,9 -1,85 0,75 1 -1,2 -1,15 -1 0 1 0,75 1 -0,95 -0,9 1 1,25 -1,15 -1,1 1 1,25 -1,8 -1,75 0 a-_a-_c+_{+ A} 0 X a-a-c+ a-_a-_c+ _a-_a-_c+_{+ A} X 0 a-_a-_c+ _a-_a-_c+_{+ A} X Cubic-LC Cubic -LC + QΓ 3 + QΓ4z a) b) c) Amp. QM 2 Cubic - LC + QΓ 3

Full Potential Energy Surface LaMnO

3 -1 0 1 -2 -1,5 -1 -0,5 0 0,5 Energy [eV] -1 0 1 AFM-A FM -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 0,75 1 -1,65 -1,6 -1 0 1 0,75 1 -1,9 -1,85 0,75 1 -1,2 -1,15 -1 0 1 0,75 1 -0,95 -0,9 1 1,25 -1,15 -1,1 1 1,25 -1,8 -1,75 0 a-_a-_c+_{+ A} 0 X a-a-c+ a-_a-_c+ _a-_a-_c+_{+ A} X 0 a-_a-_c+ _a-_a-_c+_{+ A} X Cubic-LC Cubic -LC + QΓ 3 + QΓ4z a) b) c) Amp. QM 2 Cubic - LC + QΓ 3

Strain PES

-0.03 -0.015 0 0.015 -25 -12.5 0 12.5 25

∆E /fu [meV]

±0.03 -0.015 0 0.015 0.03 Amp. Q3Γ- AFM-A Q3Γ- FM Q4zΓ Q4zΓ -AFM-A -FM a) P-1 b) Pnma +Δ Q3 R +Δ Q2 M

LaMnO

3

LaNiO

3

Bilayers on SrTiO

3

Projected Band Structure

Γ X S Y Γ Z U R T Z Y T U X S R -3 -2 -1 0 1 2 3 E - E F [eV] O- p Mn-e_g Mn-t_2g La-f

Influence of U|J Parameters

AFM-A FM U=5, J=1.5 U=8, J=2 Energy [e V] 0 -0.2 -0.4 -0.6 -1 -0.5 0 0.5 ±1 -0.5 0 0.5 1 Amp. QM 2